初基演算

<正> 命题演算的构成,通常有三步骤的说法,即从 Johanson 的“极小演算”到 Heyting的构造论命题演算再到二值演算.此外,Lewis 从模态或严格蕴涵出发,也分别了许多步骤,以达到二值演算为其极限;特别值得注意的是最后三个步骤,即从 S4 到 S5 到二值演算.这两个三步骤就某意义说乃是通常命题演算的构成中最本质的步骤.综合这两个三步骤,会带来许多便利,而本文所提出的也就是作为二者共同基础的初基演算.

THE BASIC CALCULUS

The Basic Calculus may be conceived as a part of the intersection ofJohanson's Minimalkalkül and Lewis's modal calculus S4.It uses twoprimitive inference-schemas:(?)The following are its axiom-schemas(14 in number):A1 A(?)AA2(B(?)C)(?)A(?)(B(?)C)]A3 A(?)(A(?)B)(?)BA4(A(?)B)(?)(B(?)C)(?)(A(?)C)A5 A(?)B(?)AA6 A(?)B(?)BA7(A(?)B)(?)(A(?)C)(?)(A(?)B(?)C)A21 A(?)A ∨ BA22 B(?)A ∨ BA23(A(?)C)(?)(B(?)C)(?)(A∨B(?)C)A24 A(?)(B∨C)(?)(A(?)B)(?)(A(?)C)A32(A(?)B)(?)B(?)A A33 A(?)(B(?)B)A34 A(?)(A(?)B)(?)BThe Basic Calculus has two important properties:(1)Let A,B,…be any given propositions.A/B is an inference of thecalculus(i.e.,fits into a primitive or derived inference-schema of thecalculus)if and only if A(?)B is a theorem of the calculus.Further,(?)is an inference of the calculus if and only if A(?)B(?)C is a theorem ofthe calculus.Similarly for three or more premises.This property is aweakened form of the“Deduction Theorem”.(2)LetΦ(X)stand for Ψ(X,C,D,…),where Ψ is constructed purelyby combination of the four primitive connectives.Then(?)is an inference-schema of the calculus.This property corresponds to whatis called the“Regularity Theorem”.(If we leave out the three axiom-schemas A24,A33,A34,both proper-ties will still belong to the resulting Ultrabasic Calculus.)If A2 is replaced byB1 B(?)(A(?)B)we get the Minimalkalkül.If,on the other hand,we add the followingtwo axiom-schemas:C1(A∨B)(?)A(?)BE1 A(?)B∨(?)Bwe get S4.Doing both things at once gives the ordinary Two-ValuedCalculus.